Dissection sparse direct solver for indefinite finite element matrices and application to a semi-conductor problem

Transcription

1 10th FreeFem++ days 2018, 13 Dec. Dissection sparse direct solver for indefinite finite element matrices and application to a semi-conductor problem Atsushi Suzuki 1 1 Cybermedia Center, Osaka University atsushi.suzuki@cas.cmc.osaka-u.ac.jp joint work with François-Xavier Roux, ONERA, LJLL Sorbonne Université

2 Outline Dissection solver for indefinite matrix overview of nested-dissection algorithm for sparse matrix LDU factorization with symmetric pivoting computation of a solution of singular linear system extension with 2x2 pivoting and unsymmetric pivoting kernel detection algorithm semi-conductor problem mixed finite element formulation of a semi-conductor problem Newton iteration to find a solution of stationary problem example of 3D computation

3 abstract framework V : Hilbert space with inner product (, ) and norm. bilinear form a(, ) : V V R continuous : γ > 0 a(u, v) γ u v u, v V. α 1 > 0 α 2 > 0 a(u, v) sup α 1 u u V. v V,v 0 v a(u, v) sup α 2 v v V. u V,u 0 u find u V s.t. a(u, v) = F (v) v V has a unique solution. U V subspace find u U s.t. a(u, v) = F (v) v U in general, inf-sup condition in subspace U is unclear. in discretized problem : V h V? in linear solver (subspace of V h )?

4 matrix form of mixed finite element method u R Nu, p R Np [ ] [ ] [ ] [ ] u A B T u f K = = p B ɛi p g A: coercive (A x, x ) > 0 x 0 B T : satisfies discrete inf-sup condition, i.e. kerb T = { 0} ɛ > 0 Schur complement matrix S = ɛi + BA 1 B T (S p, p ) = ɛ( p, p ) + (BA 1 B T p, p ) ɛ( p, p ) > 0 LDL T -factorization is possible for any ordering we need to set appropriate regularization ɛ > 0

5 recursive» generation» of Schur» complement by nested-dissection A11 A 21 A11 0 I1 A = 1 11 A 12 A 21 A 22 A 21 S 22 0 I 2 S 22 = A 22 A 21 A 1 11 A 12 = A 22 (A 21 U 1 11 )D 1 11 L 1 11 A 12 : recursively computed 8 9 a b c d e f aa bb 5a 5b a 2b 19 cc dd 6c 6d 3d 1b 1c 1d ee ff 7e 7f 3e 3f 1e a5 b c6 d e7 f a2 b d3 d1 e3 e1 f b1 c Schur complement by sparse solver Schur complement by Schur complement by dense solver dense factorization sparse part : completely in parallel dense part : better use of BLAS 3; dgemm, dtrsm dense solver

6 Symmetric pivoting with postponing for block strategy : 1/2 nested-dissection decomposition may produce singular sub-matrix for indefinite matrix τ : given threshold for postponing, 10 2 A(i, i) / A(i 1, i 1) < τ {A(k, j)} i k,j are postponed i 1 i invertible entries candidates of null pivot Schur complement matrix from postponed pivots is computed

7 Pivoting strategy full pivoting : A = Π T L LUΠ R find max k<i,j n A(i, j) partial pivoting : A = ΠLU find max k<i n A(i, k) k k symmetric pivoting : A = Π T LDUΠ find max k<i n A(k, k) 2 2 pivoting : A = Π T L D UΠ find max k<i,j n det A(i, i) A(i, j) A(j, i) A(j, j) k k sym. pivoting is mathematically not always possible

8 Computation of a solution of singular linear system A R N N, k = dimkera. index ordering {i 1, i 2,, i N } V N k = span[ e i1, e i2,..., e in k ] assumption 1 : V N k kera = { 0} A 1 11 Solution in the space V N k x = A f x VN k (A x f, v) = 0 v V N k» A 1 N 1 = 11 A» 12 N I 2 = A T 11 AT 21, kera = spann 2 I 1, kera T = sapnn 2 2 assumption 2 : u = A f is computable by LDU-factorization with symmetric pivot. f ImA f KerA T N T 2 f = 0 A 21 A 1 11 f 1 f 2 = 0 A A f = f ζ R k, A( u + N 1 ζ) = f u + N 1 ζ (KerA) N T 1 ( u + N 1 ζ) = 0 : algebraic inverse u + N 1 ζ VN k ζ = 0 : computational u + N 1 ζ ImA N T 2 ( u + N 1 ζ) = 0 ImA KerA = { 0} ImA KerA = { 0} N T 2 N 1 is invertible N 1 η ImA KerA N T 2 N 1 η = 0, ImA KerA = { 0} ( N T 2 N 1 η = 0 η = 0 )

9 extension with 2x2 pivoting or unsymmetric pivoting For matrix from PDE, most of part can be factorized with symmetric pivoting. ga 11 has LDU-factorization with symmetric pivot entries of S postponed pivots + some invertible entries of A g A ga 11»»» A11 A A = 12 A11 0 I11 A = 11 1 A 12 A 21 A 22 A 21 S 0 I 2 S = A 22 A 21 A 1 11 A 12 applying full pivot when S is (highly) unsymmetric, S = Π L SΠ R applying 2x2 pivot when S is symmetric + indefinite with full pivoting Π L /Π R S = Π L SΠ R =» S22 S23 S 32 S33 =» E 33 = 0 Ker S = span S 1 S I 3 2 KerA = 6 4»» S22 0 I2 S 1 S S 32 E 33 0 I 3» S 1, KerS = spanπ S R I 3 3 A 1 11 A(2) 12 A 1 11 A(3)» 12 I (2) 7 S Π S R I (3) I 3 2 S

10 novel kernel detection algorithm based on LDU A : N N unsymmetric, dimkera = k 1, dimima m. two parameters: l, n, which define size of factorization,»»» N n A11 A 12 A11 0 I1 A n A 21 A = A 21 S A12 gim 22 0 I n = span 2»Ā 1 11 A12 I 2. projection : P n : R N g Im n solution in subspace, Ā N l b =»Ā 1 11 b 1 0» b1, b = b2 N l l : computed in quadruple-precision with perturbation to simulate double-precision round-off error. kernel detection algorithm DOI: /nme.4729 n : candidate of dimension of the kernel compute for l = n 1, n, n + 1 err (n) l := max Pn max (Ā N la x x), max Ā N la x x x=[ 0 x l ] 0 x x=[ x N l 0] 0 x Ā 1 11 n = k + 1 err n 1 0 err n 0 err n+1 1 n = k err n 1 0 err n 0 err n+1 1 n = k 1 err n 1 0 err n 0 err n+1 1 Theoretically A 1 N k+1, but err(k) k 1 is computable; Ā 1 N AN IN.

11 Solution of singular system in FreeFem++ : 1/2 Navier-Stokes equations with full Dirichlet boundary conditions load "Dissection" defaulttodissection(); solve NavierStokes([u1,u2,p],[v1,v2,q], solver=sparsesolver,tolpivot=1.0e-2) = int2d(th)( [u1,u2] *[v1,v2]*rho/dt + UgradV(u1,u2,up1,up2) *[v1,v2]*rho + 2*nu *(Eps(u1,u2):Eps(v1,v2)) - p * div(v1,v2) - q *div(u1,u2)) // - eps*p*q ) - int2d(th)( [up1,up2] *[v1,v2]*rho/dt - rho*v2 ) +on(1,u1=0,u2=0) without penalization for indefinite system without penalization for Dirichlet boundary condition by tgv

12 Solution of singular system in FreeFem++ : 2/2 Accessing to kernel vectors in cavity driven flow problem load "Dissection" defaulttodissection(); varf vdns ([u1,u2,p],[v1,v2,q]) = int2d(th)( + nu * ( dx(u1)*dx(v1) + dy(u1)*dy(v1) + dx(u2)*dx(v2) + dy(u2)*dy(v2) ) + p*dx(v1)+ p*dy(v2) - dx(u1)*q- dy(u2)*q + (Ugrad(u1,u2,up1,up2) *[v1,v2] + + Ugrad(up1,up2,u1,u2) *[v1,v2]) + on(1,2,3,4,u1=1,u2=1) matrix Ans=vDNS(XXMh,XXMh,tgv=1e+30); real[int] b = vns(0,xxmh,tgv=1e+30); real[int,int] kernn(1,1), kernt(1,1); // N x kerndim int kerndim; set(ans,solver=sparsesolver,strategy=2, tolpivot=1.0e-2); dissectionkernel(ans,kerneldim=kerndim, kerneln=kernn,kernelt=kernt); cout << "kern dim = " << kerndim << endl; Thanks to Pierre Jolivet, GetSolver() added in MarticeCreuse.pp, github: develop

13 Numerical simulation of semi-conductor problem governing equation non-linear system of Drift-Diffusion equations discretization scheme to satisfy conservation law FVM : Scharfetter-Gummel method, 1969 mixed FEM + hybridization + mass lumping = FVM M-matrix property Brezzi-Marini-Pietra, 1987 nonlinear solver Gummel map (a kind of fixed point method), 1964 full Newton iteration linear solver to tread large condition number Bi-CGSTAB : van der Vorst, 1992 Pardiso : Schenk-Gärtner-Schmidthüsen-Fichtner, 1999 mixed formulation with H(div) L 2 directly written by FreeFem++ indefinite unsymmetric system is solved by Dissection stably.

14 Drift-Diffusion system at stationary state ϕ : electrostatic potential n : electron concentration p : hole concentration div(εe) = q(p n + C(x)) divj n = 0 divj p = 0 q : positive electron charge ε : dielectric constant of the materials E = ϕ D n = µ n θ, D p = µ p θ : Einstein s relation J n = q(µ n n ϕ D n n) J p = q(µ p p ϕ + D p p) Maxwell-Boltzmann statistics : p=n i exp( ϕ p ϕ )leads to θ J p = qµ p p ϕ p ϕ p : quasi-fermi level N i : intrinsic concentration of the semiconductor θ = k T/q : thermal voltage k : Boltzmann constant, T : lattice temperature

15 Drift-Diffusion system at stationary state : 2/3 dimensionless system (by De Mari scaling or unit scaling) div(λ 2 ϕ) = p n + C(x) divj n = 0 divj p = 0 J n = n n ϕ J p = ( p + p ϕ) with boundary conditions ϕ = f on Γ D n = g on Γ D p = h on Γ D ϕ ν = 0 on Γ N J n ν = 0 on Γ N n ν = 0 J p ν = 0 on Γ N p ν = 0

16 Boundary conditions for a diode device (thanks to Dr. Sho ) (4) (3) (5) (6) (2) N region : x < 1/4 y > 3/4, C(x, y) = nd = 1017 P region : others C(x, y) = na = 1020 n p = n2i, ni = charge neutrality p n + C(x, y) = 0 on (1), (4). N region p: n = n2i + C 2 /4 + C/2 ' nd P region p: p = n2i + C 2 /4 C/2 ' na (1) (1) (2), (3), (5), (6) (4) ϕ = log( nnai ) + 1 Vth ϕappl ν ϕ = 0 ϕ = log( nndi ) ni na ν n = 0 nd n= ni n= na ni ν p = 0 ni p= nd p=

17 Finite volume discretization with Scharfetter-Gummel method Slotboom variables, ξ : p = ξe ϕ p = ξ(e ϕ ) ξe ϕ ϕ = e ϕ ξ p ϕ J p = p p ϕ = e ϕ ξ approximation of e ϕ J p = ξ in an interval [x i, x i+1 ], h = x i+1 x i xi+1 x i e ϕ dx J p i+1/2 = h ξ i+1/2 (ξ i+1 ξ i ) ϕ : assumed to be linear in the interval [x i, x i+1 ] with ϕ i and ϕ i+1 xi+1 e ϕ dx = 1 [e ϕ(x)] x i+1 h = (e ϕi+1 e ϕi ) dϕ x i x i ϕ d x i+1 ϕ i J p i+1/2 (ξ i+1 ξ i ) ϕ i+1 ϕ i 1 h e ϕi+1 e ϕi = ϕ i+1 ϕ i e ϕi+1 p i+1 e ϕi p i h e ϕi+1 e ϕi = 1 h (B(ϕ i ϕ i+1 )p i+1 B(ϕ i+1 ϕ i )p i ) B(x) = x/(e x 1) : Bernoulli function

18 Mixed variational formulation : 1 / 2 Slotboom variable ξ : p = ξe ϕ div(j p ) = 0 in, e ϕ J p = ξ in. function space : H(div) = {v L 2 () 2 ; div v L 2 ()}, Σ = {v H(div) ; v ν = 0 on Γ N } integration by parts leads to e ϕ J p v = ξ v = ξ v ξv ν e ϕ J p v ξ v = he ϕ v ν Γ D ξv ν Γ N J p q = 0 q L 2 () v Σ

19 Mixed variational formulation : 2 / 2 mixed-type weak formulation find (J p, ξ) Σ L 2 () e ϕ J p v ξ v = he ϕ v ν Γ D J p q = 0 replacing ξ = e ϕ p again, v Σ q L 2 () symmetric indefinite find (J p, p) Σ L 2 () e ϕ J p v e ϕ p v = he ϕ v ν Γ D J p q = 0 v Σ q L 2 () unsymmetric indefinite hybridization of mixed formulation + mass lumping FVM

20 FreeFem++ script Finite element approximation J p H(div) : Raviar-Thomas : p L 2 () ϕ H 1 () : piecewise linear : P 1(K), : piecewise linear : P 1(K). RT 0(K) = (P 0(K)) 2 + xp 0(K), load "Dissection" defaulttodissection(); fespace Vh(Th, RT0); fespace Ph(Th, P1); fespace Xh(Th, P1); Vh [up1, up2], [v1, v2]; Ph pp, q; Xh phi; // obtained in Gummel map problem DDp([up1, up2, pp],[v1, v2, q], solver=sparsesolver,strategy=2, tolpivot=1.0e-2,tgv=1.0e+30) = int2d(th,qft=qf5pt)(exp(phi) * (up1 * v1 + up2 * v2) - exp(phi) * pp * (dx(v1) + dy(v2)) + q * (dx(up1) + dy(up2))) +int1d(th, 1)(gp1 * exp(phi) * (v1 * N.x + v2 * N.y)) +int1d(th, 4)(gp4 * exp(phi) * (v1 * N.x + v2 * N.y)) + on(2, 3, 5, 6, up1 = 0, up2 = 0);

21 Numerical result of a semi-conductor problem compute thermal equilibrium with p = n i exp( ϕ)v th and n = n i exp(ϕ)v th Newton iteration for the potential equation and fixed point iteration for the whole system : a kind of Gummel map electrostatic potential hole concentration

22 Raviart-Thomas finite element RT 0(K) = (P 0(K)) 2 + xp 0(K) (P 1(K)) 2. K : triangle element {E i } : edges of K K E i ν i : outer normal of K on E i Ei : normal to edge E i given by whole triangulation v RT 0(K) v Ei n i P 0(E i ), P i div v P 0(K) finite element basis E i Ψ i ( x) = σ i 2 K ( x P i ) σ i = E i ν i, P i : node of K finite element vector value is continuous on the middle point on E i. inner finite element approximation of H(div ; ). e ϕ h Ψj Ψ i e ϕ1λ1+ϕ2λ2+ϕ3λ3 λ k λ l by exact quadrature K K e ϕ h Ψj Ψ i 1 e P k ϕ kλ k Ψ j K Ψ i : exponential fitting K {λ 1, λ 2, λ 3 } : barycentric coordinates of K K K E i ν i

23 Numerical comparison on accuracy of quadrature relative error of each entry of the stiffness matrix m i j = during 4 iterations of the Gummel map #nnz=31,249 # 1 # 2 quadrature 1% max 1% max qf5pt qf7pt qf9pt # 3 # 4 quadrature 1% max 1% max qf5pt qf7pt qf9pt K e ϕ h Ψj Ψ i ϕ h e ϕ h

24 Nonlinear iteration to obtain the stationary state a variant of Gummel map ϕ m, n m, p m : given ϕ m+1, n m+1, p m+1 by a fixed point method Slotboom variable ξ m, η m p m = ξ m e ϕm n m = η m e ϕm to find a solution of the nonlinear eq. : div(λ 2 ϕ) = p n + C(x) F (η m, ξ m ; ϕ, ψ) = λ 2 ϕ ψ (ξ m e ϕ η m e ϕ + C)ψ = 0 differential calculus with δϕ {H 1 () ; ψ = 0 on Γ D } leads to F (η m, ξ m ; ϕ m + δϕ, ψ) F (η m, ξ m ; ϕ m, ψ) ( = λ 2 δϕ ψ ξ m e ϕm δϕ e ϕm) ( ψ η m e ϕm +δϕ e ϕm) ψ ( = λ 2 δϕ ψ + ξ m e ϕm + η m e ϕm) δϕ ψ = λ 2 δϕ ψ + (p m + n m )δϕ ψ ϕ m+1 = ϕ m + δϕ

25 mixed-type weak formulation find (ϕ, J n, n, J p, p) H 1 () H(div; ) L 2 () H(div; ) L 2 () λ 2 ϕ ψ = (p n + C)ψ ψ H 1 (), e ϕ J n v + e ϕ n v = g n e ϕ v ν v H(div; ), Γ D J n q = 0 q L 2 (), e ϕ J p v e ϕ p v = g p e ϕ v ν v H(div; ), Γ D J p q = 0 q L 2 (). Newton method + initial guess given by a solution by Gummel map coding by FreeFem++ and Dissection 1st order : P1/RT0/P1/RT0/P1 instead of P1/RT0/P0/RT0/P0

26 Newton iteration 1st order scheme twice repeated iteration of Gummel map produces a good initial guess for Newton iteration

27 3D computation : 1/2 tetrahedral mesh with refinement by tetgen # nodes = 29,107 # elements = 165,163 1st order : P1/RT0/P1/RT0/P1 DOF for ϕ = 27,942, (J n, n) = 354,756, full Newton = 73,7454

28 3D computation : 2/2 computing thermal equilibrium by Newton iteration for ϕ Gummel map : fixed point iteration for (J n, n) and (J p, p) + Newton iteration for ϕ, 3 iterations full Newton iteration after 3 Gammel iterations rhs 1e-06 1e-08 1e-10 1e-12 1e iteration Dissection solver for the 4th Newton iteration postponed entries = 8 with τ = 10 2 error = e-09, residual = e-16

29 Summary Full pivoting procedure works as a preconditioner for kernel detection algorithm based on symmetric pivoting Kernel detection routine in Dissection is accessible from FreeFem++, github develop tangent matrix of Newton method with high condition number for highly nonlinear semi-conductor problem is well factorized by Dissection ongoing merging in FreeFem++ ver. 4 quadruple precision solution from given matrix data by double precision with inner iterative refinement by Dissection source code of Dissection is accessible within FreeFem++ repository master/download/dissection under GPL linking-exception / CeCILL-C licenses